An asymptotic preserving scheme for the defocusing Davey–Stewartson II equation in the semiclassical limit

Abstract

This article is devoted to constructing an asymptotic preserving method for the defocusing Davey–Stewartson II equation in the semiclassical limit. First, we introduce the Wentzel–Kramers–Brillouin ansatz $\varPsi =A^\varepsilon e^{i\phi ^\varepsilon /\varepsilon }$ for the equation and obtain the new system for both $A^\varepsilon $ and $\phi ^\varepsilon $, where the complex-valued amplitude function $A^\varepsilon $ can avoid automatically the singularity of the quantum potential in vacuum. Secondly, we prove the local existence of the solutions of the new system for $t\in [0,T]$, and show that the solutions of the new system are convergent to the limit when $\varepsilon \rightarrow 0$. Finally, we construct a second-order time-splitting Fourier spectral method for the new system and numerous numerical experiments show that the method is uniformly accurate with respect to $\varepsilon $, i.e., its accuracy does not deteriorate for vanishing $\varepsilon $, and it is an asymptotic preserving one. However, it might not be uniformly convergent.